INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 7 (2005) R53–R72 doi:10.1088/1464-4266/7/5/R02

REVIEW ARTICLE

Spatiotemporal optical solitonsBoris A Malomed1, Dumitru Mihalache2, Frank Wise3

and Lluis Torner4

1 Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty ofEngineering, Tel Aviv University, Tel Aviv 69978, Israel2 Department of Theoretical Physics, Institute of Atomic Physics, PO Box MG-6, Bucharest,Romania3 Department of Applied Physics, 212 Clark Hall, Cornell University, Ithaca, NY 14853, USA4 ICFO-Institut de Ciencies Fotoniques, and Department of Signal Theory andCommunications, Universitat Politecnica de Catalunya, Barcelona 08034, Spain

Received 1 December 2004, accepted for publication 1 February 2005Published 15 April 2005Online at stacks.iop.org/JOptB/7/R53

AbstractIn the course of the past several years, a new level of understanding has beenachieved about conditions for the existence, stability, and generation ofspatiotemporal optical solitons, which are nondiffracting and nondispersingwavepackets propagating in nonlinear optical media. Experimentally,effectively two-dimensional (2D) spatiotemporal solitons that overcomediffraction in one transverse spatial dimension have been created in quadraticnonlinear media. With regard to the theory, fundamentally new features oflight pulses that self-trap in one or two transverse spatial dimensions and donot spread out in time, when propagating in various optical media, werethoroughly investigated in models with various nonlinearities. Stablevorticity-carrying spatiotemporal solitons have been predicted too, in mediawith competing nonlinearities (quadratic–cubic or cubic–quintic). Thisarticle offers an up-to-date survey of experimental and theoretical results inthis field. Both achievements and outstanding difficulties are reviewed, andopen problems are highlighted. Also briefly described are recent predictionsfor stable 2D and 3D solitons in Bose–Einstein condensates supported byfull or low-dimensional optical lattices.

Keywords: Bose–Einstein condensation, collapse, dispersion management,Gross–Pitaevskii equation, group-velocity dispersion, light bullet,matter-wave soliton, modulational instability, nonlinear optics, nonlinearSchrodinger equation, optical lattice, optical vortex, second-harmonicgeneration, self-focusing, solitary wave, spatial soliton, spatiotemporalsoliton, variational approximation, vortex soliton

(Some figures in this article are in colour only in the electronic version)

List of acronyms used in the text

1D one dimensional2D two dimensional3D three dimensionalBC boundary conditionsBEC Bose–Einstein condensationBG Bragg grating

CQ cubic–quintic (nonlinearity)FF fundamental frequencyGVD group-velocity dispersionGVM group-velocity mismatchLB light bulletMI modulational instabilityNLS nonlinear Schrodinger (equation)OL optical lattice

1464-4266/05/050053+20$30.00 © 2005 IOP Publishing Ltd Printed in the UK R53

B A Malomed et al

SH second harmonicSHG second-harmonic generationSIT self-induced transparencySTS spatiotemporal solitonVA variational approximationVK Vakhitov–Kolokolov (stability criterion)

1. Introduction: basic concepts and terminology

A general property of electromagnetic wavepackets is that theytend to spread out as they evolve. A fundamental cause for thisis that distinct frequency components, which are superposedto create the wavepacket, propagate with different velocitiesand/or in different directions. An example is the transversespreading of a laser beam due to diffraction. Similarly,light pulses spread in time as they propagate in a materialmedium, as, due to the group-velocity dispersion (GVD), eachFourier component of the pulse has a different velocity. Theseexamples pertain to linear propagation of beams or pulses.Nonlinear effects generally accelerate the disintegration of awavepacket. However, under special conditions, nonlinearitymay compensate the linear effects. The resulting balancedlocalized pulse or beam of light, that propagates without decay,is generally known as a soliton or, more properly, a solitarywave (sometimes, the term ‘soliton’ is reserved for pulsesin exactly integrable nonlinear-wave models). Thus, opticalsolitons are localized electromagnetic waves that propagatestably in nonlinear media with dispersion, diffraction or both.

Temporal solitons in single-mode optical fibres arethe prototypical optical solitons; these were predicted in1973 (Hasegawa and Tappert 1973), and first observedexperimentally in 1980 (Mollenauer et al 1980). Theformation of such temporal solitons can be understoodintuitively as follows. In an optical fibre (or any other suitableoptical medium), the effective index of refraction n dependson the intensity I , so that n = n0 +�n(I), where, in the firstapproximation, the small nonlinear correction is proportionalto the intensity, �n(I) = n2 I . Usual materials feature self-focusing, which implies n2 > 0, and instantaneous nonlinearresponse of the medium (the latter means that there is notemporal delay between �n(I) and I ); such materials arereferred to as Kerr media. In a more general case, the mediumis self-focusing or self-defocusing in the case of, respectively,d(�n)/dI > 0 or d(�n)/dI < 0.

In the course of the propagation, the pulse accumulates aphase shift that, due to the intensity-dependent correction n2 Ito the refractive index, mimics the temporal shape of the pulse,I = I (t). To understand this in a more accurate form, one maystart from the normalized wave equation for the electric fieldE ,

Ezz − (n2 E)t t = 0, (1)

where z is the propagation distance, t is time, and n is theabove-mentioned refractive index (detailed derivation of thewave equation can be found, e.g., in the book by Agrawal(1995)). A solution to equation (1) is looked for as

E(z, t) = u(z)eik0 z−iω0 t + u∗(z)e−ik0 z+iω0 t , (2)

where exp(ik0z − iω0t) represents a rapidly varying carrierwave, and u(z, t) is a slowly varying local amplitude.

Substituting this in equation (1), in the lowest approximationone obtains a dispersion relation between the propagationconstant (wavenumber) k and the frequency ω, k2

0 = (n0ω0)2,

and the next-order approximation yields an evolution equationfor the amplitude,

idu

dz+

n0n2

k0ω2

0 Iu = 0 (3)

(actually, I = |u|2, as the intensity is tantamount to thesquared amplitude). A solution to equation (3) is simply�φ = (n0n2)(ω

20/k0)I z, where �φ is a nonlinear shift of

the wave’s phase. Because the corresponding frequency shiftis �ω = −∂�φ/∂t , one obtains

�ω = −n0n2

k0ω2

0dI

dtz. (4)

It follows from equation (4) that the lower-frequencycomponents of the pulse, with�ω < 0, develop near its front,where dI/dt > 0, while higher frequencies, with �ω > 0,develop close to the rear of the pulse, where dI/dt < 0.

Actually, the dielectric response of the material mediumis not instantaneous, but is characterized by a finite temporaldelay. This implies that the linear part, ε ≡ n2

0, of themultiplier n2 in the wave equation (1) (the dynamic dielectricpermeability) is actually a linear operator, rather than simplya multiplier. The accordingly modified form of the linear termbecomes (

∫ ∞0 ε(τ)E(t − τ) dτ)t t , where τ is the delay time.

Finally, approximating this nonlocal-in-time expression by aquasi-local expansion, ε0 Ett +ε2Etttt +· · ·, gives rise to second-and higher-order GVD terms in the eventual propagationequation, which can be translated into the corresponding lineardispersion relation, k = k(ω) (Agrawal 1995).

In particular, the normal GVD (which means that waveswith a higher frequency have a smaller group velocity, asconveniently expressed by the condition β2 ≡ d2k/dω2 > 0)would reinforce the above trend to the temporal separationbetween the low- and high-frequency components of the pulse,contributing to its rapid spread. In contrast, anomalous GVD(β2 < 0), which occurs too in real materials, may compensatethe nonlinearity-induced spreading. With the magnitudes ofthe dispersion and intensity properly matched, the balancemay be perfect, giving rise to very robust pulses, that arereferred to as (1 + 1)-dimensional ((1 + 1)D, for brevity)solitons, where the first ‘1’ corresponds to t , and the second‘1’ relates to z. Extensive research has led to the developmentof soliton-based telecommunications systems (Hasegawa andKodama 1995, Iannone et al 1998). The first commercial fibre-optic telecommunications link using the solitons (actually,they are dispersion-managed solitons, which implies thatthe local dispersion coefficient β2(z) periodically alternatesbetween positive and negative values), about 3000 km long,was launched in Australia in 2003 (McEntee 2003). Whilethis is the first direct commercial application of the solitonconcept, we note that related phenomena helped to achieveother important technological advances over the years, suchas self-focusing which led to Kerr-lens mode locking for thegeneration of short laser pulses (see, e.g., a paper by Haus(2000)).

Spatial solitons originate from a balance of self-focusingand diffraction. In this case, a laser beam experiences the

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Kerr-induced nonlinear contribution to the index of refractionthat follows the spatial (rather than temporal) profile of theintensity, and thus acts as an effective lens. In one transversedimension, the diffraction formally resembles the anomalousdispersion in the temporal domain (see above); therefore theintensity-dependent lens can exactly compensate diffraction,and the resulting beam may propagate without spreading orself-compression. Such a beam has the form of a stripe ina planar waveguide (the waveguide confines the beam in theother transverse direction), as was first observed by Aitchisonet al (1990). In view of the above-mentioned mathematicalsimilarity of the one-dimensional diffraction to anomalouschromatic dispersion, the existence of soliton solutions (whichmay also be regarded as (1 + 1)D objects, where, this time, thefirst ‘1’ refers to the transverse spatial coordinate, x , ratherthan t in the case of temporal solitons) in this case is notsurprising. However, (2 + 1)D spatial solitons (self-formedcylindrical beams) in media with the Kerr (cubic, or χ(3))nonlinearity are unstable, unlike their (1 + 1)D counterparts,because two-dimensional fluctuations may destroy the balancebetween the nonlinearity and diffraction in that case. Inparticular, an increase of the intensity leads to self-focusingof the cylindrical beam, which further increases the intensityand the corresponding intensity-dependent correction to theindex of refraction, which leads to still stronger focusing andincrease of the intensity, and so on. This self-acceleratingprocess of the nonlinear self-focusing is referred to as collapseof the beam (see, e.g., the review by Berge 1998). In anidealized mathematical model of the Kerr medium, the collapsewould result in formation of a singularity after passing a finitepropagation distance, but in reality new physical phenomenaenter into the play, such as higher-order nonlinearities, non-paraxiality, and material damage. One way to avoid such abehaviour and stabilize the (2 + 1)D spatial soliton is to havesaturation of the nonlinearity: if the nonlinear coefficient n2

itself depends on the intensity I , so that it starts to decreasewith I , the self-focusing may reach a stable equilibrium withdiffraction.

Studies of spatial solitons have made rapid progress sincethe mid-1990s, when two new soliton-supporting nonlinearoptical interactions became available to experiments (seeStegeman et al 2000). First, Segev et al (1992) had predictedthat the photorefractive effect in electro-optic materials couldbe exploited to create a saturable nonlinear index of refractionthat would support solitons. Photorefractive solitons wereobserved experimentally soon afterwards (Duree et al 1993),and since then a variety of such solitons, of both 1D and2D types, have been discovered and explored (Segev andChristodoulides 2002). This includes landmark advances, suchas self-trapping of incoherent light (Mitchel and Segev 1997)and the recent generation of ordinary (Fleischer et al 2003)and vortex (Fleischer et al 2004, Neshev et al 2004) solitons inoptically induced photonic lattices, as well as robust necklace-shaped soliton clusters (Yang et al 2005); the lattice solitons inphotorefractive media were first predicted by Efremidis et al(2002).

Second, concomitant with the work on photorefractivematerials, there was a resurgence of interest in the study ofeffects produced by the parametric interaction of two or threewaves in quadratically nonlinear (χ(2)) media. In appropriate

limits, the parametric interactions lead to a so-called cascadingmechanism, which induces an effective approximately cubic(χ(3)) nonlinearity through a superposition of two χ(2)

processes (the resulting effective χ(3) nonlinearity may beeither self-focusing or self-defocusing, corresponding to�n =n2 I with n2 > 0 and n2 < 0, respectively). This processwas identified theoretically in the 1960s, but only sporadicexperimental results appeared (Thomas and Taran 1972,Belashenkov et al 1989) until DeSalvo et al (1992) reported asystematic study of the nonlinear phase shift produced in thesecond-harmonic generation (SHG), and Stegeman et al (1993)properly recognized its potential to impress large nonlinearphase shifts on light beams and signals (for a review, seeStegeman et al 1996). This interaction becomes efficientclose to the point of phase matching between the fundamental-frequency (FF) and second-harmonic fields involved in theSHG interaction (where the phase velocities of the two wavescoincide). In this case, the cascading limit does not hold, andgenuine ‘quadratic’ parametric interactions occur.

The fact that quadratic nonlinearities feature an effectivedynamic saturation at large intensities can be readilyunderstood by noticing that the higher the intensity in onefield, the stronger the tendency to energy conversion to theother field. Hence, the exchange of energy between the twofields, and the sensitivity of the energy exchange between theFF and SH fields to the relative phase between them, are thekey ingredients of the physical picture that may stabilize thepulse or beam in this case. Thus, the presence of the twofields is a fundamental difference from the Kerr media, whichentails new experimental complications, but also opens newopportunities. Extensive work for solitons in χ(2) media hasbeen performed during the last decade (see reviews by Etrichet al (2000), Buryak et al (2002), Torner and Barthelemy(2003)). (2 + 1)D and (1 + 1)D spatial solitons in quadraticmedia were first generated experimentally by, respectively,Torruellas et al (1995) and Schiek et al (1996).

Another important example of a (2 + 1)D spatial solitonis the optical vortex, which is a soliton of the dark type, in theform of a ‘hole’ in an extended background, which is supportedby a vortex phase pattern imprinted onto the background(Swartzlander and Law 1992, Snyder et al 1992, Tikhonenkoet al 1995, Duree et al 1995, Chen et al 1997, Kivshar et al1998). This pattern is characterized by the phase circulationof 2πS, with an integer S, around the hole. Fundamentalvortices (with S = 1) are stable in the case of a self-defocusingχ(3) nonlinearity, while all the higher-order vortices withS � 2 are predicted to be unstable, splitting into a set offundamental ones. However, in χ(2) media all the vortices areunstable, as their spatially uniform background is subject tomodulational instability. Nevertheless, some complex vortexstructures, not yet fully analysed theoretically, were observedin a χ(2) medium by Di Trapani et al (2000), in cases when theinstability growth was effectively inhibited, under conditionsthat, through the above-mentioned cascading mechanism,the quadratic nonlinearity mimics a self-defocusing cubicnonlinearity for one of the waves involved in the χ(2) process.

One of the major goals in the study of nonlinear waveinteractions in optics is the generation of pulses that arelocalized in all the transverse dimensions of space, as wellas in time (for a moving soliton, the latter is equivalent to

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z

y

x

Figure 1. Illustration of the formation of a spatiotemporal solitondue to the simultaneous balance of diffraction and dispersion bynonlinear self-focusing.

localization in the longitudinal coordinate)—spatiotemporalsolitons (STS). These are (2 + 1 + 1)D objects, where the ‘2’ isthe transverse dimension, the first ‘1’ stands for the temporalvariable, and the last ‘1’ pertains to the propagation coordinate.The search for the formation of such objects dates back to theearly days of the field. In particular, such pulses in Kerr mediawere considered by Silberberg (1990), who had coined theterm ‘light bullets’ (LB) for them, which stresses their particle-like nature (a ‘light bullet’ propagating in a planar waveguidewould be a (1 + 1 + 1)D object). In contrast to the extensivedevelopments in the studies of temporal and spatial solitonsin one and two dimensions, experimental progress toward theproduction of STS in the three-dimensional case has been slow.To date, (2 + 1 + 1)D STS have not yet been observed.

The formation of (2 + 1 + 1)D solitons involves the sameconditions as for their (2+1)D and (1+1)D lower-dimensionalcounterparts, which provide for the proper balance betweenthe linear spreading and a suitable nonlinearity. Thus, STSmay be understood as the result of the simultaneous balanceof diffraction and GVD by the transverse self-focusing andnonlinear phase modulation in the longitudinal direction,respectively (figure 1). Hence, one might naively expect thatnonlinear responses suitable for the formation of a stable low-dimensional soliton would also be adequate for the formationof solitons in all dimensions. That such is not the case, and,actually, that dimensionality is a central issue in the formationof solitons (first of all, because of the problem of stabilityagainst the collapse in the multidimensional case), was learnedin the early days of the work in the field, as illustrated abovefor the case of the Kerr nonlinearity. Actually, it turns outthat an effective strength of a given nonlinearity criticallydepends on the dimensionality of the physical setting where thenonlinearity acts (see, e.g., Kuznetsov et al 1986). Therefore,progress in the area requires specific conceptual, analytical,numerical, and experimental advances. This review aims tosummarize these advances along with some important openquestions.

In summary, the quest for spatiotemporal solitons, or lightbullets, faces two main challenges: first, physically relevantmodels of nonlinear optical systems, based on evolutionequations that allow stable three-dimensional propagation,ought to be identified; second, suitable materials should befound where such models can be implemented. Thus, advancesin both theoretical and experimental directions are necessary,making it appropriate to review progress in the two areas. Wewill begin with a summary of experimental results, that willillustrate the main issues. Recent theoretical work, which bothunderlies the experiments and treats a variety of phenomenathat are still beyond current experimental capabilities, will

be summarized after that. In the presentation of theoreticalmodels and results, we chiefly focus on those models whichare closest to current experiments, i.e., the ones with the χ(2)

nonlinearity, and on issues which play the key role in theexperiments. We will also give a brief overview of theoreticalpredictions for two- and three-dimensional solitons in Bose–Einstein condensates (BECs), in the case of the self-attractivecubic nonlinearity, where the stabilization of the solitons isprovided for by optical lattices created in the BEC. Theseobjects, being very different physically from the optical STSs,nevertheless have much in common with them, as regardsthe mathematical models and understanding of the underlyingdynamics. A brief discussion of open problems is included atthe end of the review.

2. Experimental advances

2.1. Cubic nonlinear media

The following are requisites for the generation of stablecompletely localized (i.e., finite-energy) STS in homogeneousmedia: self-focusing nonlinearity, anomalous GVD, and oneor more processes that can prevent collapse of the pulse. Itis useful to define the characteristic lengths for dispersion,LDS = τ 2/|β2|, and diffraction, LDF = πρ2/λ, where ρis the radius of the beam, and λ is the wavelength of thecarrying electromagnetic wave. These are the distances overwhich a pulse/beam begins to broaden appreciably in timeand space, respectively, if the propagation is linear. Stablepassage of several characteristic lengths is required to identifya soliton in the experiment. In a medium with positive Kerrnonlinearity and anomalous GVD, a pulse will simultaneouslycompress in time and space (which is the beginning of thecollapse) if LDS ∼ LDF. Effects that are not included in thesimple model, such as multiphoton absorption and stimulatedRaman scattering, can prevent the collapse from occurring.This expectation is borne out by recent experiments. Inparticular, Eisenberg et al (2001) investigated the propagationof intense femtosecond-duration pulses in a planar fused-silicawaveguide. The wavelength was chosen so that the materialhad anomalous GVD, and the above-mentioned condition,LDS ∼ LDF (as concerns the diffraction in the waveguide’splane), held. Space–time self-focusing without collapse wasobserved for a range of input intensities (however, no solitonwas formed). Numerical simulations show that multiphotonionization and stimulated Raman scattering indeed arrest theonset of collapse in such a situation. These experimentsprovided the first direct observation of space–time focusingin a Kerr medium, but the resulting pulses are not (and cannotbe) solitons—the dissipative processes that arrest the collapsedeplete the pulse’s energy, and eventually cause it to spread outin the transverse and longitudinal (i.e., spatial and temporal)directions.

Besides the above-mentioned dissipative processes,collapse may be arrested by a variety of mechanisms thatconserve energy: non-paraxiality, higher-order dispersion(Fibich et al 2002), or self-generation of nonlocalnonlinearities mediated by the rectification, i.e., generationof a slowly varying zero-harmonic field (Torres et al 2002a,2002b, Crasovan et al 2003b, Leblond 1998a, 1998b), to

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name a few. These effects may lead to new forms of soliton-like propagation. In particular, Goorjian and Silberberg(1997) observed stable spatiotemporal propagation of few-cycle pulses over many characteristic lengths in a numericalsolution of the full system of the Maxwell’s equations in anonlinear medium.

On the other hand, Towers and Malomed (2002) hadshown that complete stabilization of a cylindrical (2 + 1)Dspatial soliton can be secured in a layered medium with‘nonlinearity management’, i.e., periodic alternation of thesign of the Kerr nonlinearity along the propagation distance.It is interesting that the same mechanism supports stable 2Dsolitons in the temporal domain—not in optics, but rather inBECs, where the periodic change of the sign of nonlinearityis provided by the so-called Feshbach resonance (Saito andUeda 2003, Abdullaev et al 2003, Montesinos et al 2004b),including the case of two-component BECs (Montesinos et al2004a). However, the model with the alternating nonlinearitysign cannot provide for stabilization of full 3D solitons.

A somewhat related theoretical result, which will bepresented in a brief form below, predicts stabilization ofpulsating STSs in a (1 + 1 + 1)D medium combining the cubicnonlinearity and the above-mentioned dispersion management,i.e., periodic alternation of the sign of the GVD coefficient asa function of the propagation coordinate z (Matuszewski et al2004). In this case too, stabilization is not possible in the full(2 + 1 + 1)D case.

2.2. Quadratic nonlinear media

Following the pioneering work by Karamzin and Sukhorukov(1976), as early as in 1981 it was shown that solutions formultidimensional solitons, including the full (2 + 1 + 1)D STS,exist and are stable in χ(2) media (Kanashov and Rubenchik1981). Malomed et al (1997) reported the first treatment ofSTS in quadratic media based on the variational approach(VA), and delineated the conditions under which STS wouldbe stable. Some details of the VA will be discussed below; itis important to mention that the theoretical prediction of STSin this case motivated the experiment.

Several of the experimental issues that must be addressedin the study of χ(2) media are consequences of the two-field nature of the problem, which involves the FF andSH components. A major impediment to the experimentalrealization of STS is the requirement of anomalous GVDat both harmonics, of a magnitude large enough that thedispersion lengths be small compared to the length of availablequadratic crystals (in fact, there is a possibility to observelong-lived soliton-like propagation in the case when theGVD is slightly normal at SH, as was shown by Torner(1999), and by Towers et al (2003), and will be discussedin some detail below). The latter is usually limited to afew centimetres, which implies that the dispersion lengthmust be below 1 cm. In many materials it is not possibleto choose the carrier wavelength so that the FF and SHsimultaneously experience anomalous GVD and negligibleabsorption. Typically, acceptably transparent materialsonly have large anomalous GVD at wavelengths in the 2–4 µm range, where sources of well-controlled high-energyfemtosecond-duration pulses, needed to launch nonlinear

xy

soliton

Figure 2. Schematic diagram of the experimental arrangement usedto observe (1 + 1 + 1)D spatiotemporal solitons.

propagation, perform poorly. A closely related issue is that,generally, the group velocities of pulses at the FF and SHdiffer substantially; the resulting group-velocity mismatch(GVM) significantly restricts the range of parameters overwhich solitons may form. In the linear propagation, the FFand SH pulses would separate temporally at a characteristicGVM propagation length (LGVM = cτ/(n1g − n2g), with c thevelocity of light, τ the pulse duration, and n1g and n2g the groupindices at the FF and SH, respectively), which tends to weakenthe coupling between the harmonics and naturally counterssoliton formation. The solitons that form despite the presenceof small GVM are referred to as ‘walking’ ones, because theyactually propagate at some velocity falling within the intervalbetween the group velocities of the FF and SH fields, i.e., they‘walk’ in the reference frame that moves at the FF groupvelocity (Torner et al 1996, Etrich et al 1997). However, thetemporal group-velocity mismatch prevents spatial trappingnear phase matching for pulses that are too short in the temporaldimension—typically the limit is a few picoseconds (Carrascoet al 2001a). The latter feature was directly confirmed in theexperiment (Pioger et al 2002).

GVD and GVM can effectively be controlled by tiltingthe amplitude fronts of a pulse with respect to the phase fronts.The tilt is impressed on a pulse through angular dispersion,most commonly by the use of prisms or diffraction gratings,and allows desired values of GVD and GVM to be obtained atwavelengths convenient to the experiment, such as 800 nm or1 µm. The ability to control both GVD and GVM with tiltedpulse fronts was exploited in the first experimental observationof temporal solitons in χ(2) media, by Di Trapani et al (1998).In that case, the tilt was used to create effective anomalousGVD for pulses launched into a barium metaborate (BBO)crystal.

Liu and co-workers used tilted pulses to produce (1 + 1 +1)D STS (Liu et al 1999, 2000a). Pulses of 120 fs durationat 800 nm were converted into a narrow stripe by means ofcylindrical optics, as shown in figure 2. The pulse tilt wasimposed in the x-direction, and the beam was focused inthe y-direction onto a crystal of lithium iodate or BBO. Inlithium iodate the tilt could be arranged so that the GVMwas close to zero, while the propagation could be monitoredover three dispersion/diffraction lengths. A pulse launchedwith suitable phase mismatch and intensity evolved to a stablespatiotemporal profile, and was observed at the exit face of thecrystal, compressed in both time and space. Roughly 10% ofthe input energy was radiated away, owing to the fact that thelaunched pulse contained the FF only and had thus to rearrange

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(a) (b)

Figure 3. Numerical simulations that show the formation of a (1 + 1 + 1)D spatiotemporal soliton. Temporal (upper panels) and spatial(lower panels) profiles of the fundamental (left) and harmonic (right) are shown.

-800 -600 -400 -200 0 200 400 600 800

Time Delay (fs)

117 fs

280 fs

108 fs

τ = 104 fs

(GW/cm )2

12

10

3

input

Intensity

Figure 4. Experimental observations of (1 + 1 + 1)D spatiotemporal solitons and modulational instability of larger pulses. The spatialprofiles on the left and temporal profiles on the right were recorded with the corresponding input intensities listed in the centre of the figure.At low intensity (3 GW cm−2), the beam diffracts in the vertical direction and the pulse disperses in time. At 10 GW cm−2, the field at theoutput face of the crystal (∼5 characteristic lengths) supports its compressed shape in time and space and is very close to the launched field.At 12 GW cm−2 (when the pulse’s energy is larger than it must be in the soliton), the beam breaks into elliptical filaments, which graduallyevolve into circular ones. The temporal profile is that of a single filament.

itself into the two-wave soliton. These pulses were the firstexperimentally observed optical spatiotemporal solitons, aswell as the first temporal solitons in χ(2) media to be directlyobserved over several characteristic lengths.

Walking STS were studied in experiments with BBO,in which the material dispersion and birefringence allowedLDS/LGVM ∼ 3 (Liu et al 2000a). This relation impliesthat, in the linear propagation, the FF and SH pulses wouldmove apart by ∼3 times the initial pulse duration whilepassing one dispersion length. The large GVM reduces theacceptance range for the soliton formation; experimentally,solitons could only be observed with the phase mismatchgreater than a threshold value, in agreement with the theory(Carrasco et al 2001b). With the phase mismatch near theoptimum value, as determined by numerical simulations (seefigure 3), stable spatiotemporal propagation was observed overfive characteristic lengths. The measured spatial and temporalprofiles are shown in figure 4. A series of similar experimentswith varying phase mismatch and intensity allowed Liu et alto delineate threshold conditions for the STS formation, whichagree well with numerical simulations.

One of the limitations to soliton formation is modulationinstability (MI). Only the (2 + 1 + 1)D STS is truly a stableobject; all lower-dimensional solitons in the 3D medium aresubject to MI (Kanashov and Rubenchik 1981), and hencethey tend to break up in time and/or space. Starting fromthe conditions that produce STS, an increase of the inputintensity by ∼20% causes the stripe to spontaneously breakinto filaments along the major axis, see figure 4 (Liu et al2000b). Higher intensities produce denser filaments, which isa clear signature of the MI. The filaments are initially elliptical,but quickly make their cross section circular, while maintaining∼100 fs temporal duration. Numerical simulations show thatit should eventually be possible to generate true LBs throughthis instability of the (1 + 1 + 1)D STS. However, the angulardispersion used to control GVD and GVM in the experimentsof Liu et al caused the filaments to disperse soon after theirformation, thus precluding the generation of fully confinedSTS.

Finally, we briefly mention that (1 + 1 + 1)D STS canalso be generated through a non-collinear interaction of twoinput pulses. Through mutual nonlinear trapping of the input

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FF fields and the SH field, STS form and propagate along thebisector of the angle between the input beams (Liu et al 2000c).This Y-geometry was originally suggested by Drummond et al(1999), and naturally lends itself to implementation of thelogical AND operation: a pulse is detected along the bisectoronly if both input pulses are present.

The use of tilted pulse fronts is an undesirable featureof all STS produced experimentally to date, as it precludesthe generation of (2 + 1 + 1)D light bullets. Thus, newstrategies for the potential generation of STS without pulse tiltare required. Some recent theoretical efforts in this directionwill be described below.

3. Basic theoretical models

3.1. Single-component models

It is well known that, within the framework of thestandard approach based on slowly varying amplitudes ofthe electromagnetic field and paraxial approximation fordiffraction, the evolution of the local field amplitude u ina dispersive multidimensional nonlinear medium (which isintroduced similarly to equation (2)) is governed by thegeneralized nonlinear Schrodinger (NLS) equation (Newelland Moloney 1992, Akhmediev and Ankiewicz 1997, Kivsharand Agrawal 2003):

iuz + (1/2)(∇2⊥u + σuττ ) + f (|u|2)u = 0; (5)

cf equation (3). Here, as above, z is the coordinate alongthe propagation direction of the carrier wave, τ ≡ t − z/Vgr

is the ‘reduced time’, with Vgr being the group velocity ofthe carrier wave (the reduced time has the same meaning asin fibre optics (Agrawal 1995)), σ is the GVD coefficient,which depends on the wavelength of the carrier wave (actually,it is tantamount to −β2 which was defined above), ∇2⊥ isthe Laplacian (diffraction operator) acting on the transversecoordinates (x, y), and the function f (|u|2) describes anonlinear correction to the effective refractive index of thematerial medium. As explained above in the context ofequation (3), the classical Kerr effect corresponds to f (|u|2) =K |u|2, where K is a constant. We denote the cases with theself-focusing and self-defocusing cubic nonlinearity, i.e., K >

0 and K < 0, as χ(3)+ and χ(3)− , respectively. The above-mentioned model of the medium composed of alternating self-focusing and self-defocusing layers, which supports stable (2+1)D spatial solitons, corresponds to K (z) periodically jumpingbetween positive and negative values (Towers and Malomed2002). In accordance with what was also described above, thecases with σ > 0 and σ < 0 in equation (5) are referred to asanomalous and normal GVD. In what follows below, we willonly consider the case of the anomalous dispersion; therefore,if σ is constant, we may always set use the normalizationσ ≡ +1 (in the single-component model). On the other hand,the dispersion management (in the multidimensional medium)corresponds to σ(z) periodically varying between positive andnegative values (Matuszewski et al 2004, 2005). Even if σ andthe coefficients of the nonlinearity, f (|u|2), are functions of z,equation (5) conserves two dynamical invariants, namely, the

norm (which is the power, in the spatial-domain models, andenergy, in the spatiotemporal-domain ones), and momentum,

E =∫

|u|2 dr, P = i

2

∫(u∇u∗ − u∗∇u) dr. (6)

Note that these expressions do not depend on the form off (|u|2) in equation (5).

The first issue that needs to be taken care of in thetheoretical approach to STS is the above-mentioned wavecollapse (spontaneous formation of a singularity): solitary-wave solutions have a chance to be stable in a uniform mediumonly if the nonlinearity adopted in the model does not leadto collapse, in a given spatial dimension. In particular, LBssupported by the usual χ(3)+ nonlinearity are unstable due tothis; therefore the simple spatially uniform Kerr nonlinearityis not an appropriate setting for the creation of STS.

More accurately, the axially symmetric soliton in the 2DNLS equation (it is commonly called a Townes soliton, whichwas, as a matter of fact, the first example of a soliton consideredin nonlinear optics; see the seminal paper by Chiao et al (1964))is weakly unstable: no eigenmode of small perturbationsaround the Townes soliton is an exponentially growing one,but there is an algebraically growing mode, corresponding tononlinear instability. In contrast to this, spherically symmetricsolitons in the 3D NLS equation are strongly (linearly) unstablein the ordinary sense, featuring an unstable eigenvalue inthe spectrum of perturbation eigenmodes. Accordingly, thecollapse is weak in the 2D NLS equation: to initiate thecollapse, the energy E of the initial pulse (see equation (6))must exceed a minimum (threshold) value, which is exactlythe energy of the Townes soliton; in the 3D case, the collapsehas no threshold.

On the other hand, LBs in the χ(3)+ medium may bestabilized by creating a waveguiding channel in it, as wasshown numerically by Raghavan and Agrawal (2000). Thechannel is a cylindrical region with an increased value of therefractive index. The possibility of the stabilization of LBs ina channel is not surprising, as the usual temporal soliton in anoptical fibre may be considered as a ‘light bullet’ propagating ina strongly confining waveguide, as was stressed by Manassahet al (1988).

The simplest possibility for producing a stable STS in auniform medium is to choose a saturable nonlinearity, withf (|u|2) = −(1 + |u|2/u2

0)−1 in equation (5), where u0 is

a constant. Indeed, simulations performed by Edmundsonand Enns (1992) and Enns and Rangnekar (1992) had shownthat (2 + 1 + 1)D solitons are stable in this case. In fact,LBs supported by saturable nonlinearity are bistable objects(two different stable solitons may have equal energies), whichmakes it also possible to consider switching between them(Enns et al 1992). However, while some physical settings,for instance vapours of alkali metals, may be modelledby a saturable nonlinearity, and soliton-like objects wereexperimentally observed in them (Tikhonenko et al 1996),these media exhibit strong dissipation too.

As was shown numerically by Quiroga-Teixeiro et al(1999), and Desyatnikov et al (2000) (see further referencesbelow), another model that can generate stable higher-dimensional solitons is the one with χ(3)+ : χ(5)− , alias cubic–quintic (CQ), nonlinearity. In this case, the function f (|u|2) in

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equation (5) combines a self-focusing term, +|u|2, and a self-defocusing one, −|u|4. Coefficients in front of these terms canbe normalized so that equation (5) takes a parameter-free form,

iuz + (1/2)(∇2⊥u + uττ ) + (|u|2 − |u|4)u = 0. (7)

A nonlinearity of the CQ type was derived in some simplified(truncated) models, e.g., those combining interaction of lightwith two-level atoms and dipole interactions between the atoms(Bowden et al 1991), or the same light–matter interactionand Kerr nonlinearity of waveguides (Akozbek and John1998). Recently, experimental observations of a nonlineardielectric response in chalcogenide glasses (Smektala et al2000, Boudebs et al 2003, Ogusu et al 2004) and in organicmaterials (Zhan et al 2002), which might be modelled bythe CQ terms, in combination with additional ones (includingthose accounting for the two-photon (nonlinear) absorption),were reported. The real applicability of the CQ model to thesenewly investigated media requires additional experimentalexplorations. In particular, a recent analysis by Chenet al (2004) shows that the nonlinear loss in the availablechalcogenide glasses leaves open a window of opportunity forobserving (1+1+1)D STS, as well as (2+1)D spatial solitons,but not the fully localized (2 + 1 + 1)D ‘light bullets’.

Lastly, it is relevant to mention that multidimensionalBose–Einstein condensates (BECs) obey (with a veryhigh accuracy) the Gross–Pitaevskii (GP) equation, whosenormalized form is (Dalfovo et al 1999):

iut + (1/2)∇2u + g|u|2u + U (r)u = 0, (8)

where u(t, r) is the normalized single-atom wavefunction,g = +1 and −1 for the attracting and repelling atoms,respectively, and U (r) is the external potential (in this case,the evolution variable is time, rather than z). In the case of theBEC, the nonlinearity is always cubic. Experiments with BECalways demand the presence of a parabolic trap, correspondingto U = (1/2) 2r 2, with a constant . For the predictionof stable multidimensional solitons, crucially important isanother potential term, which accounts for a periodic opticallattice (OL), that can be easily built in the experiment as aninterference pattern induced by counter-propagating coherentlaser beams illuminating the condensate. In the 3D case, theOL potential is

U (r) = −ε[cos(kx) + cos(ky) + cos(kz)], (9)

with constant ε > 0 and k (the sign minus in front of ε impliesthat x = y = z = 0 is a local minimum of the potential).The 2D case corresponds to the expression (9) without theterm cos(kz). An interesting possibility is also presented bythe low-dimensional potential, i.e., the 2D potential in the 3Dcase, and the 1D potential in the 2D case (Baizakov et al 2004a,2004b). The low-dimensional potential is quite relevant to theexperiment, simply because it is easier to create it by means ofthe laser beams.

3.2. Two-wave models with the quadratic nonlinearity

As explained in the previous section, the only media in whichSTS have been already observed experimentally are opticalcrystals featuring the χ(2) nonlinearity; hence it is necessary to

give a review of main theoretical results obtained in respectivemodels. Equations governing the evolution of amplitudes uand v of the FF and SH fields in dispersive χ(2) bulk media,including the χ(3) nonlinearity too, are well known (Menyuket al 1994, Karpierz 1995, Buryak et al 1995, Bang et al 1998,Etrich et al 2000, Buryak et al 2002)

iuz + (1/2)(∇2⊥u + uττ ) + u∗ v − �(|u|2 + 2|v|2)u = 0, (10)

i(vz + cvτ ) + (1/4)(∇2⊥v + δ · uττ )

− βv + u2 − 2�(2|u|2 + |v|2)v = 0. (11)

Here, c and β are differences (mismatches) of the group andphase velocities between the FF and SH waves (the formerwas called GVM in the previous section), � is the χ(3)

coefficient (in this notation, � > 0 corresponds to the self-defocusing, χ(3)− , cubic nonlinearity), while the χ(2) coefficientis normalized to be 1, and δ is the ratio of the GVD coefficientsat SH and FF. Note that the GVD coefficient in equation (10)is set to be +1, which implies that the temporal dispersionat the FF is anomalous; cf equation (5) (otherwise, there isno chance to produce STS). In fact, equations (10) and (11)present the simplest model for the light propagation in mediawith competing nonlinearities, the χ(2) part of the modelcorresponding to the type-I interaction, which involves onlyone polarization of light.

Equations describing planar χ(2) waveguides differ fromthe above ones in that ∇2

⊥ is replaced by ∂2/∂x2, where x isthe single transverse coordinate. As for the (2 + 1)D spatialsolitons, equations generating them differ from the above onesby dropping the τ -derivatives. Note that, on replacing thetemporal variable τ by the second transverse coordinate y, anysingle-component model for (1 + 1 + 1)D solitons in a planarwaveguide with anomalous GVD is made formally similar toa model for (2+ 1)D spatial solitons in a bulk medium with thesame nonlinearity; therefore solitary-wave solutions may beinterpreted in either way. However, the spatiotemporal modelis exactly equivalent to the spatial one in the bulk mediumsolely if δ = 1 in equation (11) (the case of a ‘spatiotemporalisotropy’; see also below); otherwise the spatiotemporal modelis a more general one.

Equations (10) and (11) conserve the energy (or ‘numberof quanta’, also called the Manley–Rowe invariant),

E =∫ +∞

−∞dτ

∫ +∞

−∞dx(|u|2 + |v|2), or

E =∫ +∞

−∞dτ

∫ +∞

02πr dr(|u|2 + |v|2), (12)

in the (1 + 1 + 1)D and (2 + 1 + 1) cases, respectively. Besidesthe energy, other conserved quantities are the three componentsof the momentum (cf equation (12)), the Hamiltonian H , and,in the absence of the walk-off term, the z-component L of theorbital angular momentum too (Akhmediev and Ankiewicz1997).

3.3. Other models

A combination of the χ(2) nonlinearity and linear dispersioninduced by the resonant reflection on a Bragg grating (BG)in a 3D medium gives rise to another collapse-free modelwhich, in principle, can support STS, as was shown by Heand Drummond (1998). The corresponding model is based

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on a system of equations for the amplitudes u± and v± of theright- and left-travelling FF and SH waves,

i(C−1∂t − ∂x + ∇2⊥)u+ + u∗

+v+ + λu− = 0,

i(C−1∂t + ∂x + ∇2⊥)u− + u∗

−v− + λu+ = 0,

2i(∂t − ∂x + ∇2⊥ − β)v+ + u2

+/2 + v− = 0,

2i(∂t + ∂x + ∇2⊥ − β)v− + u2

−/2 + v+ = 0.

(13)

Here, C and λ are the group velocity and Bragg reflectivity ofthe FF wave, while the same coefficients for SH are normalizedto be 1, andβ is the phase mismatch; cf equations (10) and (11).Effectively stable STS were found in simulations of this model.However, one should take into regard that the linear spectrumof this system contains no true bandgap, the presence of whichis a necessary condition for the existence of solitons in therigorous sense. Therefore, the STS found in this system mayactually have tiny nonvanishing ‘tail’, similarly to the STSfound by Towers et al (2003) in the χ(2) model with relativelyweak normal GVD at the SH.

STS were also studied theoretically in other systems,such as models of off-resonance two-level media (Mel’nikovet al 2000), and the Maxwell–Bloch equations describing self-induced transparency (SIT) in a multidimensional medium,

−i∇2⊥E + n2Eτ + Ez + i(1 − n2)E − P = 0, (14)

Pτ − EW = 0, Wτ + (1/2)(E∗ P + P∗E) = 0, (15)

where E and P are the complex electric field and polarizationof the medium, real W is the population inversion of the two-level atoms, and n is the refractive index. Equations (14)and (15) do not give rise to collapse; therefore LBs may existand be stable in this model, as was shown numerically byBlaauboer et al (2000). Besides that, stable LBs in the SITmedium exist in a guiding channel (Blaauboer et al 2002).The channel can be created by giving the refractive index n(see equation (14)) a graded profile in the radial direction,similar to the above-mentioned one, proposed by Raghavanand Agrawal (2000) to stabilize LBs in the χ(3)+ medium.

Coming back to the χ(2) nonlinearity, another potentialstrategy for achieving formation of fully three-dimensionalLBs was suggested by Torner et al (2001). Their schemeis based on the concept of tandem structures, which arecomposed of periodically alternating linear dispersive and χ(2)

nonlinear layers, so that the nonlinearity and GVD are providedby different materials. In this scheme, the STS are composedof guiding-centre wavepackets that ought to periodically passpieces with high quadratic nonlinearity but no GVD, followedby linear pieces with GVD only. Numerical simulationsshow that long-lived STS can form in a variety of tandemstructures with feasible domain lengths (Torner et al 2001).A step forward along this direction has been reported recentlyby Carrasco et al (2004), who experimentally observed self-focusing of light in tandems with periodically varying Poyntingvector walk-off.

4. Main theoretical results

As was mentioned above, theoretical results for LBs inmedia with various nonlinearities are, currently, far more

advanced than experimental observations. The objective of thissection is to summarize the most essential theoretical results,which should suggest perspectives for further advances in theexperimental studies.

4.1. Solitary-wave solutions

A solution to the 3D NLS equation (5) (with the GVDparameter σ = 1) describing a stationary STS (generallyspeaking, a ‘spinning’ one) is sought in the form

u(z, r, θ, τ) = U (r, τ) exp(iκz + iSθ), (16)

where r and θ are the polar coordinates in the transverse plane,real κ is a wavenumber, S = 0, 1, 2 . . . is the vorticity (‘spin’),and the real function U (r, τ) obeys the following boundaryconditions (BC):

U (r, τ) ∼ ξ−(D−1)/2 exp(−√

2κξ), ξ 2 ≡ r 2 + τ 2 (17)

for r, |τ | → ∞ (D = 2 and 3 correspond to the (1 + 1 + 1)Dand (2 + 1 + 1)D cases, respectively; in the former case,ξ 2 ≡ x2 + τ 2), and U ∼ r |S| when r → 0 (if S = 0, theBC at r = 0 is Ur (r = 0) = 0). A corollary of equation (17)is that solitons may only exist with κ > 0.

The form (16) implies that the spin (if S = 0) is alignedwith the propagation direction z. Solutions of a more generaltype would be extremely complex; therefore they have neverbeen considered (in a system of two nonlinearly coupled 3Ddiscrete NLS equations with the cubic nonlinearity, a stablecomplex of two orthogonal vortices was recently found byKevrekidis et al (2004b); that model has no interpretation interms of the guided-wave propagation, but applies to BECs andlattices of coupled resonant microcavities).

Similar to equation (16), a stationary-soliton solution toequations (10) and (11) is sought for as

u(z, r, θ, τ) = U (r, θ) exp(iκz + iSθ),

v(z, r, θ, τ) = V (r, θ) exp(2iκz + 2iSθ)(18)

(the functions U and V may be assumed real if the GVMparameter c is zero). These stationary solutions constituteone-parameter families of solitons, with the wavenumber shiftκ acting as an intrinsic parameter of the family (in the presenceof the walk-off, the families of stationary walking solitons arecharacterized by a second parameter, the soliton’s velocity).The stationary STS realize extrema of the Hamiltonian H atfixed energy E : δF(H + κE) = 0, where δF stands for thefunctional, or Frechet, derivative.

These STS exist for values of wavenumber shifts κ suchthat the soliton is not in resonance with the linear dispersivewaves, to avoid energy leakage from the soliton. This conditionconfines an existence domain of the solitons. Once this domainwas found, stability of solitons is a crucial issue. In the case ofsingle-parameter (non-walking) families of solitons, a linear-stability border is determined by the Vakhitov–Kolokolov(VK) criterion, dE/dκ = 0 (Vakhitov and Kolokolov 1973).This result can be understood in terms of simple geometricalarguments, recalling that, as was said above, the stationarysolitary wave solutions realize a local extremum (maximumor minimum) of H for given energy E (see, e.g., Kuznetsov

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0 5000 100000.00

0.05

0.10

0.15

S=1S=2

S=0

wav

enu

mbe

r

energy

0 5000 10000-1600

-800

0

S=2

S=1

S=0

(b)(a)

Ham

ilton

ian

energy

Figure 5. The wavenumber κ (a) and Hamiltonian H (b) of the 3Dsolitons in media with focusing cubic and defocusing quinticnonlinearities versus their energy E .

et al 1986, Akhmediev and Ankiewicz 1997, Torner et al1995a). The lower branch of the E–H diagram (where theE–H curve is concave) may represent a stable soliton family,while the solitons situated on the upper branch (where theE–H curve is convex) are all definitely unstable. When thelowest-order soliton solutions realize an absolute minimum ofH for given E , then they are stable (for a recent overview ofthe use of Hamiltonian-versus-energy curves in the analysis ofthe existence and stability of solitons in conservative systems,see Ankiewicz and Akhmediev (2003)). Typical examplesof κ = κ(E) and H = H(E) plots are shown in figure 5for the case of 3D solitons in media with focusing cubic anddefocusing quintic nonlinearities (Mihalache et al 2002a); thefull and dashed lines in this figure correspond to stable andunstable solitons (for a detailed discussion, see below thesection devoted to spinning solitons).

It should be stressed that the VK criterion does not provideinformation about possible oscillatory instability, accountedfor by pairs or quartets of complex-conjugate eigenvalues (see,for example, Akhmediev et al 1993, De Rossi et al 1998,Mihalache et al 1998b, Schollmann et al 1999). Therefore, thiscriterion has to be used with caution when applied to stationarysolutions that do not realize the absolute minimum of H forgiven E , for example those corresponding to spinning solitons.

4.2. Two- and three-dimensional solitons in models with thecubic nonlinearity

Before proceeding to non-Kerr nonlinearities, it is relevantto briefly describe the possibility of stabilization of (1 + 1 +1)D STSs in the dispersion-management χ(3) model, whichamounts to equation (5) with σ = σ(z), f (|u|2) ≡ |u|2,and one transverse coordinate x . Matuszewski et al (2004)have demonstrated this possibility using semi-analytical andnumerical methods. The analytical approach is based on theVA (variational approximation), with the following ansatz forthe soliton:

u(z, x, τ) = A(z) exp

{

iφ(z) − 1

2

[x2

W 2(z)+

τ 2

T 2(z)

]

+i

2

[b(z)x2 + β(z)τ 2]

}

, (19)

where A and φ are the amplitude and phase of the soliton,W and T are its transverse and temporal widths, and b andβ are the spatial and temporal chirps. All these variationalparameters are real. The application of the VA (as described

Figure 6. A series of snapshots, taken from direct simulations of the(1 + 1 + 1)D equation (5) with the dispersion management, whichillustrate stable oscillations of the pulse deep inside its stabilityregion.

Figure 7. The same as figure 6, but for a pulse taken close to aborder of the stability region. In this case, the spatiotemporal solitonperiodically splits into two subpulses that recombine again.

in the review by Malomed (2002)) leads to the followingevolution equations for the parameters:

b = W ′/W, β = σ−1T ′/T , (20)

W ′′ = 1

W 3− E

2W 2T, (21)

T ′′ − σ ′

σT ′ = σ 2

T 3− σ E

2W T 2, (22)

where the prime stands for d/dz, and E is the soliton’s energy.Numerical solution of these equations, followed by directsimulations of the underlying NLS equation, have made itpossible to identify a sufficiently large stability region in thesoliton’s parameter space. Direct simulations of equation (5)confirm the stability of the pulsating solitons in that region.A typical example of such a soliton is shown in figure 6.The oscillations produce no tangible radiation loss. More

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Review: Spatiotemporal optical solitons

accurate examination of the numerical data shows that someamount of radiation is emitted from the soliton when it ispassing a layer with the normal GVD, but this radiationwave is reabsorbed by the soliton within the anomalous-GVDlayer. An interesting situation occurs close to borders of thestability region: in that case, the pulse periodically splits andrecombines, remaining a stable state; see figure 7. However, inthe 3D case (corresponding to equation (5) with two transversecoordinates) neither the VA nor direct simulations can produceany stable soliton in the dispersion-managed model.

Another possibility for stabilizing both 2D and 3D solitonsin the NLS equation with the cubic self-focusing nonlinearityis provided by the OL potential (see equation (9)), as wasdemonstrated by Baizakov et al (2003, 2004a, 2004b). Inthe 2D case, the stationary solution may be approximated bythe ansatz

u = A exp[−iµt − (ax2 + by2)

]. (23)

With the full OL potential, which involves both cos(kx) andcos(ky) in equation (9), one can set a = b, while with thequasi-1D OL, which involves only cos(kx), the variationalparameters a and b must be kept independent in the formulasgiven below (the OL wavenumber is normalized to be k =2√

2, as in the works by Baizakov et al (2003, 2004a, 2004b)).The final prediction of the VA is that, with the full (two-dimensional) OL, the 2D solitons exist and are stable, pursuantto the VK criterion, in an interval

8π(1 − 8e−2ε) < E < 8π, (24)

and with the quasi-1D lattice, their existence and stabilityinterval is

8π√

1 − 8e−2ε < E < 8π, (25)

assuming ε < e2/8 ≈ 0.924 (the norm E , defined as perequation (6), has the meaning of the number of atoms in thecondensate, rather than the energy in the optical models). Thepurport of these results is that, in the absence of the OL (ε = 0),both intervals (24) and (25) shrink to a single point, E = 8π ,which is nothing but the VA prediction for the norm of theTownes soliton (a numerically found value of its norm is 23.4;i.e., the VA predicts it with an error of �7%). Thus, the singlevalue of the norm of the weakly unstable (neutrally stable, interms of its eigenvalue spectrum) Townes soliton in the free 2Dspace is stretched by the OL into a finite interval, (24) or (25),in which the emerging soliton family is stable.

Direct numerical simulations corroborate these predic-tions quite well. Figure 8 shows examples of tightly andloosely self-trapped 2D solitons supported by the quasi-1Dpotential. The predictions (24) and (25) were also found to bein a reasonable agreement with numerical results. The forma-tion of solitons in the 3D lattice potential is, generally, similarto the 2D case, although the relaxation time is shorter, dueto the stronger interaction involving a larger number of adja-cent cells. A typical example of a stable multi-cell 3D soliton,which forms itself in a strong lattice potential, is given in fig-ure 9(a).

Additionally, in the full 2D lattice, stable solitons withembedded vorticity were found, despite the fact that thevorticity is not a dynamical invariant in the presence of the

2.0

1.5

1.0

0.5

0.0-10 0 10

x

-10

0

10

0.8

0.6

0.4

0.2

0.0-10 0 10

x

-10

0

10

y

y

(a)

(b)

Figure 8. Typical examples of stable single-humped (tightly bound,(a)) and multi-humped (loosely bound, (b)) two-dimensionalsolitons in the Gross–Pitaevskii equation with attraction, supportedby the quasi-one-dimensional optical lattice.

2

1

0-10 -5 0 5 10

-10

-5

0

5

10

y

x

2

2

2

1

0-6 0

d6

2

6420

-6 -4 -2 0 2 4 6

-6-4

-20

24

6

x

y

(a)

(b)

Figure 9. (a) An established 3D soliton formed in a strong latticepotential. (b) An example of a two-dimensional vortex soliton withS = 1, supported by the square lattice. The vorticity is defined asS ≡ �φ/(2π), where �φ is a change of the phase of the complexstationary wavefunction along a closed path surrounding the centralpoint. The inset additionally shows the diagonal cross section of thevortex.

OL (see an example in figure 9(b)). In the 3D case, boththe full 3D lattice and its quasi-2D counterpart support stable3D solitons (Baizakov et al 2003, 2004a, 2004b), in contrastto the model with the ac-Feshbach-resonance modulation ofthe nonlinearity, which could not stabilize 3D solitons (Saito

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and Ueda 2003, Abdullaev et al 2003). It is worth notingthat the quasi-1D lattice cannot stabilize 3D solitons, or2D vortex solitons (Baizakov et al 2004a, 2004b); however,a combination of the quasi-1D OL with the ac-Feshbachresonance gives rise to stable solitons in the 3D system(Matuszewski et al 2005).

A very recent result, reported by Kartashov et al (2004a,2004b) is the existence of various stable solitons in the2D medium equipped with a radial lattice, in the formcorresponding to an OL that, as is well known (Durnin et al1987), can be created by a diffraction-free Bessel beam.Moreover, the same 2D radial lattice may also support stable3D solitons (Mihalache et al 2004c). The simplest version ofthe corresponding GP equation (8) in the 3D case is

i∂u

∂t= −1

2

(

∇2⊥u +

∂2u

∂z2

)

− |u|2u − AJ0

(r

r0

)

u,

where J0 is the Bessel function, with constant A and r0,(actually, the potential ∼ J0 in this equation assumes amixture of a weak Bessel beam with a strong spatially uniformbackground (Mihalache et al 2004c)) and ∇2

⊥ is the transverseLaplacian acting on the coordinates x and y, with r ≡√

x2 + y2.The above BEC models, except for the one with the full

3D lattice (but including the radial Bessel lattice), also finddirect interpretations in terms of nonlinear optics. Indeed, ift is realized as the propagation distance z, then the 3D modelwith the quasi-2D lattice applies to the propagation of light in abulk Kerr (self-focusing) medium with the periodic transversemodulation of the refractive index in x and y. Thus, stable LBscan be predicted in such a setting. Indeed, stable one-parameterfamilies of 3D spatiotemporal solitons in self-focusing cubicKerr-type optical media with an imprinted 2D photonic latticehave been recently found by Mihalache et al (2004b). These2D photonic lattices can support stable 3D solitons providedthat their energy is within a certain interval and the strengthof the lattice potential, which is proportional to the refractiveindex modulation depth, is above a certain threshold value(Mihalache et al 2004b). Remarkably, for sufficiently largevalues of the lattice strength parameter, the Hamiltonian–energy curves display two cusps, instead of a single one as inother 2D and 3D Hamiltonian systems (see, e.g., Akhmedievand Ankiewicz 1997). This unique feature is intimately relatedto the existence of stable 3D solitons within a finite interval oftheir energies. This two-cusp structure is actually an exampleof the so-called ‘swallow’s-tail’ structure, which is one ofgeneric patterns known in the catastrophe theory. Actually,this pattern occurs quite rarely in physical applications (fora review on the catastrophe theory in the application to thesoliton stability see, e.g., Kusmartsev (1989)).

Finally we stress that the 2D model with the 2D latticepertains to spatial (rather than spatiotemporal) solitons in thesame bulk medium. Further, the 2D model equipped with thequasi-1D lattice admits two different interpretations in termsof optics. If the coordinate y along the free (unmodulated)direction is replaced by the reduced time τ , the latter modeldescribes the (1 + 1 + 1)D spatiotemporal solitons in a planarwaveguide with the refractive index periodically modulated inthe transverse direction x . If, instead, y remains the secondtransverse coordinate, then one may consider the model as

describing (1 + 1 + 1)D spatial solitons in the bulk medium,where the refractive index is modulated only in one transversedirection, x .

4.3. Spatiotemporal solitons in the model with the quadraticnonlinearity

4.3.1. Spatiotemporal solitons versus collapse. Long beforethe χ(2) solitons were observed experimentally, Karamzinand Sukhorukov (1976) had shown that stable 2D solitonsexist in quadratic media. Later, Kanashov and Rubenchik(1981) proved, by using rigorous variational estimates, thatin the 3D medium of this type, the only stable object maybe a fully localized soliton, i.e., an STS of the (2 + 1 + 1)Dtype (their analysis did not consider the role of the phasemismatch). An implication of this result is that spatial(2+1)D cylindrical solitons are, strictly speaking, unstable too,against time-dependent modulational perturbations breakingtheir uniformity along the propagation axis; from this point ofview, 1D and 2D solitons in infinitely long samples are onlytransient objects. As was explained above, the modulationalinstability of (1+1+1)D solitons in the bulk χ(2) was observedby Liu et al (2000b); see figure 4. The onset of an instability ofthis type was studied in numerical simulations of theχ(2) modelby De Rossi et al (1997), and the modulational instability gainin a (1 + 1)D system (a planar waveguide) was experimentallymeasured by Schiek et al (2001).

4.3.2. The variational approximation versus direct numericalresults. While multidimensional solitons in the χ(2) modelshould be looked for in a numerical form, useful quasi-analytical approximations can be applied too. In particular, anapproximation based on a product ansatz for the soliton wasproposed by Hayata and Koshiba (1993). A more accurateapproach, based on the VA, was developed by Malomed et al(1997) (VA was earlier applied to STS in the model with asaturable nonlinearity by Enns and Rangnekar (1993)).

VA starts with the choice of an ansatz (trial function) forthe stationary solutions U and V ; cf equations (19) and (23).As well as the above ansatze, a tractable one is based onthe Gaussian form of the fields U and V in both spatial andtemporal variables:

U = A exp(−ar2 − ρτ 2), V = B exp(−br2 − στ 2).

(26)Here, A, B and a, ρ, b, σ are variational parameters, namely,amplitudes and inverse squared spatial and temporal widths ofthe FF and SH components of the STS, and r is the transversecoordinate in the (1 + 1 + 1)D case, or the transverse radialvariable in the (2 + 1 + 1)D case.

Another ingredient of VA is the Lagrangian Lcorresponding to the stationary equations obtained from thedynamical system (10) and (11). In the case of c = 0 and realU and V , and without the χ(3) terms (� = 0), the Lagrangianis L = ∫ +∞

−∞ dx∫ +∞−∞ dτ L or L = ∫ ∞

0 r dr∫ +∞−∞ dτ L in the

(1 + 1 + 1)D and (2 + 1 + 1)D cases, respectively, where theLagrangian density is

L = 1

2(∇⊥U )2 +

1

8(∇⊥V )2 + U 2

τ /2 +δ

8V 2τ

+ κU 2 +

(

κ +β

2

)

V 2 − U 2V ≡ H + κE, (27)

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(a) (b) (c)

Figure 10. Normalized threshold intensity (a), power (b), and energy (c) of the solitons in the second-harmonic-generation model versus thenormalized phase-velocity mismatch.

H and E being the corresponding Hamiltonian and energydensities.

The Gaussian ansatz is inserted into the Lagrangian,and the integration is performed explicitly, which yields areduced Lagrangian as a function of the variational parameters.Finally, the equations that these parameters obey are derivedby equating to zero the derivatives of the effective Lagrangianwith respect to them. As was mentioned above, the VA mayalso predict, in a part, stability of the solitons, using the VKcriterion.

The VA has produced the following predictions for the(1 + 1 + 1)D and (2 + 1 + 1)D solitons in the χ(2) model:

(i) For all values δ > 0 of the relative-dispersion parameter,there is exactly one stable soliton, in both the (1 + 1 + 1)Dand (2 + 1 + 1)D cases.

(ii) There is a finite minimum energy Ethr (threshold)necessary for the existence of the solitons.

(iii) In the (1 + 1 + 1)D case, exactly one solution is predictedfor δ < 0 too, i.e., for the case when the GVD is normalat SH.

(iv) In the (2 + 1 + 1)D case, solutions for δ < 0 exist only in avery narrow stripe, e.g., for δ > −0.034 if β = −3/2. Inthis region, VA produces two different physical solutions,only one of which may be stable.

The above VA predictions are to be compared to resultsof direct simulations of the STS dynamics in the χ(2) model(Malomed et al 1997, Skryabin and Firth 1998, Mihalacheet al 1998a, 1999a). The output of numerical calculationsis illustrated in figure 10, where we plot the thresholdcharacteristics for the one-, two-, and three-dimensionalsolitons (D = 1, 2, 3), in the case of δ = 1, as functions ofthe normalized phase-velocity mismatch β = k1�

20�k, where

�k ≡ 2k1 − k2, �0 is the spatial width of the beam, andk1 and k2 are the FF and SH wavenumbers, respectively. Asimple scaling analysis shows that, for positive and negativemismatches, the threshold behaves as |β|(4−D)/2. It is observedthat, for the positive mismatch, there is no threshold forthe formation of (1 + 1)D solitons; moreover, the thresholdvanishes at the exact phase-matching point (β = 0) for anydimension D. Remarkably, for spatial solitons (D = 2) andat the positive mismatch, an exact expression holds (Torneret al 1995b): Pthr = βPNLS, where PNLS � 5.85 is the knownnormalized collapse threshold of the 2D NLS equation (it isthe same as the norm of the Townes soliton which was quotedabove in a different normalization).

Comparison with numerical simulations shows that theabove predictions (i) and (ii) are correct, aside from the fact

(a) (b)

Figure 11. Regions of existence and stability of walkingspatiotemporal solitons at negative (a) and positive (b) values of thephase mismatch.

that the numerically found threshold energy Ethr vanishes atthe exact-matching point, β = 0; see figure 10 (the lattercircumstance is missed by VA).

The features (iii) and (iv), i.e., the existence ornonexistence of STS in the case when the dispersion is normalat SH, deserve special discussion. This issue is a physicallysignificant one because, as was explained in detail above, inavailable χ(2) materials it is quite difficult to find a case whenGVD is anomalous at both FF and SH simultaneously, withoutstrong dissipation.

Initial calculations performed by Mihalache et al (1998a)did not produce STS in the case when the GVD at SH wasnormal. However, semi-analytical and numerical studiesreported by Towers et al (2003) suggest that there is a well-defined region in the parameter plane (β, δ), where stablespatiotemporal soliton-like pulses, of both (1 + 1 + 1)D and(2 + 1 + 1)D types, exist. In the rigorous sense, those pulsesmay not be solitons, as their SH component may have anattached ‘tail’, which does not vanish at infinity. Nevertheless,numerical computations show that, for small enough normalGVD, the tail’s amplitude, if any, is smaller than the peakamplitude of the pulse by a factor of (at least) ∼104, and hencethe pulses may be identified as solitons in the experiment.A related issue, which was investigated by Beckwitt et al(2003), is the possibility of the existence of stable temporal(1D) solitons in the same case, with the result that an effectiveexistence border is found at still larger values of the normalGVD at SH. Notice that the existence of long-lived soliton-likeobjects at small and even at moderate normal GVD at the SH,with anomalous GVD at the FF, was also predicted by Torner(1999). These results indicate the possibility of experimentalsearch for STS-like objects under such conditions.

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The effects of the GVM on STS were considered in detailby Mihalache et al (1999b) for the (1 + 1 + 1)D case, andby Mihalache et al (2000c) for the (2 + 1 + 1)D case. One-parameter and two-parameter families of walking STS werefound to exist, most of them being stable. These familiesof solitons are natural generalizations of 1D and 2D walkingsolitons introduced by Torner et al (1996), Etrich et al (1997),and Mihalache et al (1997). As in the spatial case, it was foundthat, in most cases, the stationary STS with the temporal walk-off continue to exist and remain stable, if they were stable atzero GVD (c = 0). This feature is illustrated by figure 11 forc = 1 and δ = 0.5, which demonstrates that most of the two-parameter families of the 3D walking solitons, characterized bythe corresponding soliton velocity and nonlinear wavenumbershift, are stable. The walking solitons are unstable only ina narrow strip between the dotted lines in figure 11 and thesolid lines which delineate their existence border (notice theasymmetry of the stability domains with respect to the sign ofthe phase mismatch β).

The effect of the temporal GVM is more conspicuous interms of the threshold energy Ethr, necessary for the existenceof the STS: Ethr no longer vanishes at the zero-phase-mismatchpoint (β = 0) if c = 0. In the absence of GVM, simple scalingarguments predict that Ethr must be proportional to |β|1/2 forboth β > 0 and β < 0. In the case of δ < 0, a maximumvalue of GVM was found by Towers et al (2003), up to whichthe soliton-like STS, found for normal GVD at SH, continuesto exist.

To summarize, whole families of stable STS exist abovethe threshold value of the light intensity, and almost allmembers of the soliton families are stable. However, this doesnot guarantee that it is easy to generate the solitons; in fact,suitable conditions for their generation still must be found.For example, for a given material and input light conditions,increase of the input intensity does not necessarily improveefficiency of the soliton generation (Torner et al 1999, Carrascoet al 2002). In any case, the key step for the generation of theLBs is identification of suitable materials, pump wavelengthsand pulse conditions, for which all the necessary conditions,as concerns GVD, GVM, losses, etc, are fulfilled.

4.4. Spinning solitons

A distinct species of multidimensional solitons is the one withintrinsic vorticity (often referred to as ‘spin’). Unlike theabove-mentioned vortex solitons on square lattices, in isotropicmedia the spin is related to the angular momentum, whichis a dynamical invariant. Both (2 + 1)D spatial solitons inan isotropic bulk medium and (2 + 1 + 1)D STS may carrythe spin. Finding a stationary shape of spinning solitons isnot difficult, but their stability is an issue. Spinning spatialsolitons in models containing a single type of the nonlinearity,such as quadratic or saturable, are strongly unstable againstazimuthal perturbations breaking the axial symmetry (Kruglovet al 1992, Torner and Petrov 1997, Firth and Skryabin 1997,Torres et al 1998). As a result, they split into sets of stablezero-spin solitons, which move away, so that the initial spinmomentum of the unstable soliton is converted into the orbitalmomentum of the splinters. The spontaneous instability of thespinning (2+1)D spatial solitons was observed experimentally

in both saturable media (Tikhonenko et al 1995, Bigelow et al2004) and in quadratic nonlinear media (Petrov et al 1998).When the instability is externally induced, it can be used toimplement the concept of soliton algebra (Minardi et al 2001).In these experiments, vorticity was impressed on the inputbeams by means of holographic phase masks. Similarly, aspinning (2 + 1 + 1)D STS in the χ(2) model was numericallyfound to be unstable against azimuthal perturbations, that alsosplits it into a set of separating zero-spin solitons (Mihalacheet al 2000b).

As was inferred, in a general form, by Mihalacheet al (2002a), spinning STS may be stabilized if the modelincorporates competing (self-focusing and self-defocusing)nonlinearities. Examples are the above-mentioned CQ model,see equation (7), and the quadratic–cubic one, based onequations (10) and (11) with � > 0. In this context, it isrelevant to mention that the strength of the effective cubicnonlinearity in χ(2) crystals may be tuned by means of opticalrectification (generation of a nearly dc field component, asshown by Torres et al (2002a, 2002b)).

For the first time, the existence of stable spinning spatial(2 + 1)D solitons with spin S = 1 in the CQ model wasdemonstrated by means of direct simulations by Quiroga-Teixeiro and Michinel (1997). Later, the issue was studied indetail by means of accurate numerical computations of linear-stability eigenvalues for the corresponding stationary solitonsolutions (Towers et al 2001b, Malomed et al 2002). Theexistence of stable spinning spatial solitons (also referred toas ‘ring vortices’) in the quadratic–cubic medium was shownnumerically, within the framework of the same approach, byTowers et al (2001a). In both models, stable spinning solitonsshare common features: they have relatively large stabilityregions for S = 1 and 2, and much narrower ones for S � 3(Pego and Warchall 2002, Mihalache et al 2004d, 2004e,Davydova and Yakimenko 2004).

In the latter context, it is relevant to mention that, in the 2Ddiscrete NLS equation, stable vortex soliton may have S = 1(Malomed and Kevrekidis 2001) and S = 3 (Kevrekidis et al2004a), while the discrete solitons with S = 2 are unstable(but stable quadrupole solitons can be found in lieu of them).Additionally, S = 1 vortices with a phase shift π betweenthem (i.e., opposite signs of the wave fields) can form stablebound states on the discrete lattice (Kevrekidis et al 2001).

The stability of the spinning (2+1+1)D solitons in the CQmodel was investigated in detail by Mihalache et al (2002a)(the stationary shape of these solitary waves can be foundnot only in a numerical form (Mihalache et al 2000a), butalso in an approximate semi-analytical one by means of VA,which is based on the ansatz U (r, τ) = V (r) sech(µτ), as wasshown by Desyatnikov et al (2000)). The soliton has a toroidal(‘doughnut-like’) shape; see figure 12. The one-parameterfamily of STS can be characterized by the dependence betweenthe wavenumber κ and energy; see equations (16) and (12).Noteworthy features of STS in CQ nonlinear media can besummarized as follows:

(i) There is an energy threshold necessary for the theirexistence, which steeply increases with the spin (seefigure 5).

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(a) (b)

Figure 12. Recovery of a stable spatiotemporal soliton with S = 1 propagating in media with competing nonlinearities, which was stronglyperturbed at input. (a) Isosurface plots of the initial perturbed soliton; (b) the self-cleaned output soliton.

(ii) In the case of S = 0, the VK criterion is sufficient forthe stability. However, for the spinning solitons it doesnot guarantee stability against perturbations destroyingthe axial symmetry of the ‘doughnut’; see further detailsbelow.

(iii) The solutions exist in a finite interval of the wavenumbers,0 < κ < κmax = 3/16, the limit of κ → κmax

corresponding to E → ∞ (see figure 5). The divergenceof the energy in this limit takes place via swelling of theouter radius of the ‘doughnut’, while the amplitude ofthe field remains limited. At the point κ = κmax the STSgoes over into an infinitely extended background field withembedded vorticity and a hole in the centre, i.e., a darkvortex soliton.

As was stated above, stability is the central issue in thestudy of the spinning solitons. Qualitative information aboutthe stability is provided by numerical propagation of solutionsin the presence of random perturbations, and quantitativeinformation is provided by numerical computation of thecorresponding eigenvalues for small perturbations obeying theequations linearized about the stationary solution. In the CQmodel, a general solution for the perturbation eigenmodes canbe sought for in a self-consistent form,

u(Z , r, T , θ)− U (r, τ) exp [i(Sθ + κz)]

= fn(r, τ) exp {λnz + i[(S + n)θ + κz]}+ g∗

n(r, τ) exp{λ∗

nz + i[(S − n)θ + κz]}, (28)

where n > 0 is an integer azimuthal index of the eigenmode( fn, gn), and λn is its instability growth rate. The self-consistency of the expression (28) means that the twoazimuthal harmonics exp[i(S + n)θ ] and exp[i(S − n)θ ] forma closed system. Numerical calculations show that the mostpersistent unstable eigenmode has n = 2, for both S = 1and 2. With the increase of κ , the S = 1 soliton becomesstable at κ = κst ≈ 0.13, and the stability region extends upto κmax = 3/16, which limits the existence region of the STS.In other words, stable spinning solitons must have a large sizeand large energy.

Direct simulations completely corroborate the stabilityof the (2 + 1 + 1)D solitons with S = 0 and 1 in the CQmodel in the region where they are predicted to be stable.Moreover, a stable STS with S = 1 can easily self-trap frominitial configurations of a rather arbitrary shape (in particular,from the one containing a large random-noise component)with embedded vorticity, provided that the energy is large

enough. In figure 12 we show a typical self-cleaning processof the initially strongly perturbed stable soliton with S =1. The existence of stable spinning STS in optical mediawith competing nonlinearities is a generic feature: stable 3Dspinning solitons with vorticity S = 1 supported by competingquadratic and cubic nonlinearities were found to exist too(Mihalache et al 2002b). Stable 2D and 3D two-componentsolitons of the same type (with S = 1) were also found in abimodal CQ system (Mihalache et al 2002c, 2003b).

It is relevant to mention that dissipative systems, describedby the complex CQ Ginzburg–Landau equation, support stable2D spinning solitons too (Crasovan et al 2001a, 2001b).However, as is always the case with solitons in dissipativemodels, these ones do not form a continuous family. Instead,for given parameters of the equation, just two soliton solutionsexist, one stable and one unstable.

The conservative model with the CQ nonlinearity wasrecently also demonstrated to support extremely stable patternsin the form of circular clusters of solitons (also called‘necklaces’; see also below). The necklaces can carry theangular momentum (Mihalache et al 2003a, 2004a). However,it remains unknown whether a narrow stability region exists forspinning STS with higher values of the vorticity, S > 1.

A recent experimental advance towards the observationof stable spinning STS is the stable propagation of spatiallylocalized optical vortices with topological charge S = 1 andS = 2 in a self-focusing photorefractive strontium bariumniobate (SBN) crystal (Neshev et al 2004, Fleischer et al 2004).These 2D vortices are created by self-trapping of partiallyincoherent light carrying a phase dislocation, and they canbe stabilized when the spatial incoherence of light exceedsa certain threshold value (Jeng et al 2004). However, thesevortices were created on top of a virtual lattice induced by asquare-lattice grid of transverse laser beams illuminating thesample. Thus, the system is far from being isotropic, andmay be compared to the GP equation with the square OLconsidered above. Stable necklace-shaped soliton clusters inthe photorefractive medium equipped with the photonic latticewere also observed (and modelled theoretically) very recentlyby Yang et al (2005).

To conclude the consideration of spinning solitons, wenotice that the prediction of the existence of stable 3D spinningsolitons with a large size, and, accordingly, large energy inoptical media with competing nonlinearities is, as a matterof fact, the second example (after Skyrmions, providing for

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a classical-field model of nucleons) of spinning 3D localizedobjects known in physics.

4.5. Interactions between spatiotemporal solitons

Collisions and other forms of interaction between STS are animportant issue, but little has been thus far done on this topic(Stegeman and Segev 1999). However, collisions between(2 + 1 + 1)D LBs in the model with the saturable nonlinearitywere simulated by Edmundson and Enns (1993), who hadfound that the collisions seem completely elastic, with thesolitons reappearing virtually unscathed after the collision.

An analytical approach to the interaction problem ispossible if the solitons interact at a large distance, so thatoverlapping between an exponentially decaying ‘tail’ of onesoliton and the ‘body’ of the other may be treated as asmall perturbation. As was shown by Malomed (1998), aneffective potential of the interaction between well-separatedidentical zero-spin solitons with a phase differenceψ , temporalseparation T , and the distance between their centres R, is

W (�,ψ) = const · �−(D−1)/2 exp(−√

2κ�)

cosψ, (29)

where �2 ≡ R2 + T 2, and κ is the same wavenumber as inequation (16); D = 3 and 2 correspond, as before, to the(2 + 1 + 1) and (1 + 1 + 1) cases, respectively. Using thepotential (29), one can derive equations of motion for� andψ .An important peculiarity of the system is that the effective masscorresponding to the phase degree of freedom, ψ , is negative(hence local minima of the interaction potential are saddlepoints, rather than stable equilibria); see further details in thereview by Malomed (2002). In a similar way, the interactionpotential was derived by Maimistov et al (1999) for solitonsbelonging to different components in a bimodal system basedon nonlinearly coupled CQ equations; in that case, the potentialdoes not depend on the phase difference between the solitons.

5. Conclusion and open problems

Thus far, spatiotemporal solitons were investigated theoreti-cally only in basic models. Many challenging possibilitieswere not yet explored, such as spinning solitons whose spinis not aligned with the propagation direction, or anisotropicspinning solitons containing vortices with an intrinsic spatialstructure, as suggested by Freund (1999) and Molina-Terrizaet al (2001). Another interesting possibility is to search forspinning (2 + 1 + 1)D spatiotemporal solitons in the model ofa χ(2) medium equipped with the Bragg grating (the modelwas introduced by He and Drummond (1998)). A somewhatsimpler but also exciting problem is the question of stabilityof composite solitons with opposite spins (S,−S) in bimodalsystems. A very recent result by Desyatnikov et al (2005) isthat such solitons with hidden vorticity may be stable in cer-tain domains of their existence region. A similar problem wasalso considered by Leblond et al (2005) for the χ(2) : χ(3)

model, with a conclusion that hidden-vorticity solitons maybe, at least, very long lived in this case too.

Given experimental challenges in the creation of anenvironment that may support STS, a possibility that LBsmight emerge from a dynamical instability becomes attractive.In this regard, it is relevant to mention an ongoing study

of the so-called nonlinear X-waves (Di Trapani et al 2003).This is a generalization of the diffraction-free propagation ofBessel beams in a linear medium (Durnin et al 1987) to thepolychromatic case (Lu and Greenleaf 1992, Sonajalg et al1997). Linear wavepackets of the form

exp(iSθ)∫ ∞

0B(k)Jn(kr sin ζ ) exp [ik(z cos ζ − ct)] dk,

(30)where (r, θ, z) are cylindrical coordinates, S is an integer,B(k) is any smooth function, and Jn is the usual Besselfunction, undergo spatiotemporally invariant propagation (Luand Greenleaf 1992). Experimentally, such ‘X-waves’ arelaunched in the form of a conical radiation pattern with the coneangle ζ . They propagate linearly without spreading in time orspace, but, off-axis, their envelopes are only weakly localized,in contrast to the exponentially localized soliton solutions;therefore they contain infinite energy. In the experiment, themain difficulty is production of the necessary conical radiationpattern.

Di Trapani et al (2003) discovered that a quasi-localized light pattern can spontaneously emerge from unstablepropagation of a short pulse in a χ(2) crystal with normalGVD and large group-and phase-velocity mismatches. Apulse with the temporal and transverse sizes ∼200 fs and∼1 mm produced a stably propagating pattern with the sizeof 25 fs × 25 µm, which kept a few per cent of the inputenergy. This pulse exhibits characteristics of an X-wave (Trullet al 2004). Trillo et al (2002) have shown that X-waves areindeed solutions of the coupled wave equations with normalGVD and large phase mismatch. The χ(2) nonlinearity maybe helpful in this context: phase-mismatched SH generationnaturally produces conical emission, and is thus an ideal wayto launch the X-waves.

Although the nonlinear X-waves are not localizedsolutions and thus not LBs in the usual sense, they do exhibit akind of 3D spatiotemporal self-focusing and trapping. Studiesof X-waves will complement work aimed at the production oftrue STS, as nonlinear X-waves can form in systems wheresolitons cannot, e.g., when GVM is large and GVD is normal.A challenging issue that remains unexplored is a continuoustransition between true STS and X-waves. In the latterconnection, it is relevant to mention that the X-wave solutionsfor the case of the anomalous temporal dispersion, as well as fortwo-dimensional X-waves (that were not considered before),were recently reported by Christodoulides et al (2004).

In the last few years a lot of attention was devoted totransverse dynamics in nonlinear optics in cavities, namely,the study of transverse effects in passive and active opticalcavities, such as the pattern formation and localized structuresin dissipative systems (for a recent overview see Mandeland Tlidi (2004)). In such systems the prototype dynamicalequations are the complex Ginzburg–Landau and Swift–Hohenberg equations (Aranson and Kramer 2002). Thecompetition between transverse diffraction and nonlinearitiesin optical resonators leads to the formation of cavity solitonsappearing as bright spots on a homogeneous background.Recently the possibility of using these cavity solitons asself-organized micropixels in semiconductor microresonatorshas been demonstrated (Hachair et al 2004). Moreover,the formation of 3D localized structures in nonlinear optical

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resonators, confined in both the transverse plane and in thelongitudinal direction (the so-called cavity LBs), beyondthe usual mean-field limit, has been recently predicted byBrambilla et al (2004).

An important open frontier of the field is the study ofLB complexes, built from several stable solitons. Structuresinvestigated to date, such as soliton necklaces introducedby Soljacic et al (1998), Soljacic and Segev (2001), orsoliton clusters studied by Desyatnikov and Kivshar (2002),tend to expand or self-destroy in the presence of smallperturbations. However, recent numerical investigationsindicate that competing quadratic–cubic nonlinearities allowone to build more robust, yet still metastable structures(Kartashov et al 2002, Crasovan et al 2003a). As wasmentioned above, the use of the CQ nonlinearity gives rise tovery robust spatiotemporal necklaces, as shown by Mihalacheet al (2004a). Completely stable stationary necklaceclusters of spatial solitons, supported by photonic lattices inphotorefractive media, were recently created experimentallyand explained theoretically by Yang et al (2005).

Soliton formation in discrete and quasi-discrete settingshas been addressed only briefly here (for a recent reviewof the main features afforded by waveguide arrays, seeChristodoulides et al (2003)). However, this is a subjectof intense theoretical and experimental research (see, e.g.,Eisenberg et al 1998, Efremidis et al 2002, Fleischer et al 2003,Fleischer et al 2004, Martin et al 2004, Neshev et al 2004, Yangand Musslimani 2003, Musslimani and Yang 2004, Iwanowet al 2004, Xu et al 2004) motivated by potential applicationsof discrete solitons to light manipulation and control, and bynew opportunities suggested by the recent demonstration ofthe optically induced photonic lattices mentioned above. Thestudy of spatiotemporal solitons in such lattices gives rise toimportant questions open for future research.

The results obtained in optics may offer clues in the searchfor multidimensional solitons in other areas of physics. Aswas said above, the field which is closest to nonlinear opticsin terms of the theoretical description, and which has attracteda great deal of attention in the last few years, is the Bose–Einstein condensation (BEC). While only 1D solitons, bright,dark, and, very recently, ones of the gap type (Eiermann et al2004), have thus far been observed in BEC (for the ordinarydark and bright solitons in BEC, see reviews by Anglin andKetterle (2002), and by Anderson and Meystre (2002)), therecently reported theoretical results, briefly reviewed in thisarticle, suggest possibilities for the creation of stable 2Dand 3D solitons in the attractive BEC by means of opticallattices. Moreover, the use of the lattices opens a possibilityto create multi-dimensional gap solitons even in repulsiveBEC (Baizakov et al 2002). In connection with this, theexistence of spatially localized topological states of a BECwith repulsive atomic interactions confined by an optical lattice(the so-called matter-wave gap vortices) has been predictedby Baizakov et al (2004a), Sakaguchi and Malomed (2004)and Ostrovskaya and Kivshar (2004). Recently, a recipefor constructing globally linked three-dimensional structuresof topological defects hosted in trapped wave fields, suchas vortex dipoles, vortex quadrupoles, and parallel vortexrings, has been given (Crasovan et al 2002, 2003c, 2004).These vortex clusters exist as robust stationary, highly excited

collective states of nonrotating BECs and are dynamicallyand structurally stable. It is also relevant to mention thatstable ‘supervortices’, i.e., ring arrays of fundamental (S =1) vortices, with global vorticity imprinted onto the entirearray, were recently predicted by Sakaguchi and Malomed(2005) in the 2D Gross–Pitaevskii equation with the attractivenonlinearity and square lattice.

Localized X-matter waves in three dimensions formed byan ultracold Bose gas in an optical lattice, characterized by apeculiar biconical shape, have been predicted too (Conti andTrillo 2004). It is also interesting to mention a formal analogybetween the 3D optical model with the χ(2) nonlinearityand models of mixed atomic–molecular BEC. The analogysuggests a principal possibility of the existence of self-supporting soliton-like condensates in coherent mixtures ofatoms and molecules (Julienne et al 1998, Drummond et al1998, Javanainen and Mackie 1998, Cusack et al 2001).

To conclude, after many years of theoretical progress,observation of truly (2 + 1 + 1)D solitons, both spinlessand spinning, remains a great challenge. Physical settingsdescribed by suitable mathematical models need to beidentified in terms of existing materials. Spin-offs ofthe research motivated by the optical STSs have importantapplications and implications for other areas of physics.

Acknowledgments

We appreciate valuable collaborations and discussions withmany colleagues around the world. The list is too long tobe included here. The work which led to this review wassupported, in part, by the Binational (US–Israel) ScienceFoundation under grant No 1999459. BAM appreciates apartial support from the Israel Science Foundation throughgrant No 8006/03. The work of DM was supported in part byInstitucio Catalana de Recerca i Estudis Avancats (ICREA),Barcelona, Spain. FW acknowledges the support of theUS National Science Foundation under award PHY-0099564.LT acknowledges support by the Spanish Government undercontract BFM2002-02861.

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